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6.641 Electromagnetic Fields, Forces, and MotionSpring 2009

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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science

6.641 Electromagnetic Fields, Forces, and Motion

Final Exam May 23, 2006 Spring 2006 Final Exam – Tuesday, May 23, 2006, 9 AM – noon. The 6.641 Formula Sheet is attached. You are also allowed to bring three 8 ½” x 11” sheet of notes (both sides) that you prepare. Problem 1 (25 points)

y

0( 0, ) coszK x y K ky= = A sheet of surface current of infinite extent in the y and z directions is placed at x = 0 and has distribution

0( 0, ) coszK x y K ky= = . The surface current flows in the z direction. Free space with no conductivity ( 0σ = ) and magnetic permeability 0μ is present for x < 0 while for 0 < x <s a perfectly insulating medium ( 0σ = ) with magnetic permeability μ is present. The region for x > s is a grounded perfect conductor so that the magnetic field is zero for x > s. Because there are no volume currents anywhere, H χ= −∇ , where χ is the magnetic scalar potential.

0,σ μ=

σ →∞

00,σ μ=

x z

s

a) What are the boundary conditions necessary to solve for the magnetic fields for x < 0 and for 0 < x < s?

b) What are the magnetic scalar potential and magnetic field distributions for x < 0 and 0 < x < s? Hint: The algebra will be greatly reduced if you use one of the following forms of the potential for the region 0 < x < s i) sin(ky)cosh[k(x-s)] ii) cos(ky)cosh[k(x-s)]

iii) sin(ky)sinh[k(x-s)] iv) cos(ky)sinh[k(x-s)]

c) What is the surface current distribution on the x = s surface? d) Use the Maxwell Stress Tensor to find the total force, magnitude and direction, on a section

of the perfect conductor at x = s that extends over a wavelength 0< y < 2kπ and 0 < z < D?

Assume that μ in the region 0 < x < s does not depend on density so that / 0d dμ ρ = .

Hint: 2 1 1cos ( ) sin(2 )2 4

ky dy y kyk

= +∫

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Problem 2 (25 points)

Depth d ,ε σ

σ →∞→

A lossy dielectric cylindrical shell, R1 < r < R2, having dielectric permittivity ε and ohmic conductivity σ is uniformly charged at time t=0 with free volume charge density 0( 0)f tρ ρ= = . The region for 0 < r < R1 is free space with permittivity 0ε and zero conductivity ( 0σ = ). Assume that the surface charge density at 1r R= is zero for all time, 1( , )s r R t 0σ = = . The r=R2 surface is a grounded perfectly conducting cylinder so that the electric field for r>R2 is zero. The depth d of the cylinder and system is very large so that fringing fields can be neglected.

a) What is the electric field in the free space region, 0 < r < R1, as a function of time? b) What is the volume charge density and electric field within the cylindrical shell,

R1 < r < R2, as a function of radius and time? c) What is the surface charge density on the interface at r=R2? d) What is the ground current i(t)?

R1

0 , 0ε σ =

R2 i(t) 0( 0)f tρ = = ρ

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Problem 3 (25 points)

A sphere with radius Rp is comprised of a uniformly permanently magnetized material in the z direction, 0 0[ cos sin ]rzM i M i iθθ θ= − . The sphere is placed concentric within a free space spherical hole of radius R within a perfect conductor as shown in the cross-sectional drawing in the figure above. The region Rp<r< R is filled with free space with magnetic permeability 0μ . There are no volume free currents anywhere ( 0fJ = ) and there is no free surface current on the r=Rp interface.

a) Prove that the magnetic scalar potential χ obeys Laplace’s equation for 0<r<Rp and Rp<r<R where H χ= −∇ .

b) What are the boundary conditions required to determine the magnetic field in regions 0<r<Rp and Rp<r<R.

c) Find the magnetic field ( , )H r θ in regions 0<r<Rp and Rp<r<R. d) Find the free surface current density K on the r=R surface.

z

riθiθ

0μ

R

Rp0 zM M i=

σ = ∞

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Problem 4 (25 points)

I1

L

a r

0μ

( , )x tξ

,T m I2

x=0 x=L g x

A conducting string at r=0 having equilibrium tension T and mass per unit length m is stretched horizontally and fixed at two rigid supports a distance L apart. The elastic string carries a current I2 and can have transverse displacements ( , )x tξ . Another rigid wire is placed at r=a and carries a current I1. The region surrounding the string and wire is free space with magnetic permeability

0μ . Assume that transverse displacements ( , )x tξ of the string centered at r=0 depend only on position x and time t. Gravity is downwards with acceleration g.

a) To linear terms in membrane displacement ( , )x tξ , find the magnetic force per unit length on the string centered at r=0.

b) What is the governing linearized differential equation of motion of the membrane? c) What must I1 be in terms of I2, m and other relevant parameters so that the membrane is

in static equilibrium with ( , ) 0x tξ = .

d) For small membrane deflections of the form ( )ˆ( , ) Re j t kxx t e ωξ ξ −⎡ ⎤= ⎣ ⎦ find the kω −

dispersion relation. Plot the ω-k relationship showing significant intercepts on the axes and slope asymptotes. Assume that k is real and that ω can be pure real or pure imaginary.

e) What are the allowed values of k that satisfy the zero deflection boundary conditions at x=0 and x=L?

f) Under what conditions will the membrane equilibrium with ( , ) 0x tξ = first become unstable?

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